Learning Objectives

• Understand the properties of divisibility.
• Apply shortcuts to check divisibility by 3, 9, and 11.
• Analyze the relationships between multiples of numbers.
• Solve cryptarithms using patterns and reasoning.
• Explore the concept of parity in arithmetic expressions.
• Investigate the conditions under which sums and products of numbers are divisible by certain integers.
• Develop mathematical reasoning and problem-solving skills through exploration and conjecture.

Chapter Notes

Second Section

• Equations:
  • $5 + 6 - 7 + 8 = 12$
  • $5 - 6 - 7 - 8 = -16$
• Continuation:
  • Three vertical dots indicating continuation or pattern repetition.

Third Section

• Symbols:
  • Alternating plus and minus signs followed by equal signs.
  • Lines and equal signs in a similar pattern to preceding sections.
• Continuation:
  • Three vertical dots indicating continuation or pattern repetition.

Properties of Divisibility

• If $a$ is divisible by $b$, then:
  • All multiples of $a$ are divisible by $b$.
  • $a$ is divisible by all factors of $b$.
  • If $a$ divides $m$ and $n$, then $a$ divides $m + n$ and $m - n$.
  • If $a$ is divisible by both $b$ and $c$, then $a$ is divisible by the LCM of $b$ and $c$.

Divisibility by 9

• Rules:
  • If the sum of the digits of a number is divisible by 9, then the number is divisible by 9.
  • If a number is not divisible by 9, then the sum of its digits is not divisible by 9.
  • If the sum of the digits is not divisible by 9, then the number is not divisible by 9.

Examples of Divisibility

1. Find the divisibility of numbers by 9:
   • 123
   • 405
   • 8888
   • 93547
   • 358095
2. Smallest multiple of 9 with no odd digits.
3. Closest multiple of 9 to 6000.
4. Count multiples of 9 between 4300 and 4400.

Diagram Descriptions

• Diagram (i):
  • Venn diagram showing relationships between multiples of 4, 8, and 32.
  • Overlapping ovals indicating shared multiples.
• Diagram (ii):
  • Similar to (i) with different arrangements of multiples.
• Diagram (iii):
  • Concentric circles representing multiples of 32, 8, and 4.
• Diagram (iv):
  • Another arrangement of concentric circles for multiples of 4, 8, and 32.

Mathematical Reasoning

• Parity:
  • Odd ± Odd = Even
  • Even ± Even = Even
  • Odd ± Even = Odd

Cryptarithms and Problem Solving

• Example Problems:
  • GH X H = 9K
  • BYE X 6 = RAY
  • UT X 3 = PUT
  • AB X 5 = BC

Conclusion

• Understanding mathematical properties and reasoning is crucial for problem-solving and exploring various mathematical concepts.